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Linear Algebra >> Content Detail



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Readings

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All of the reading assignments below refer to the main textbook:

Amazon logo Jacob, Bill. Linear Algebra. New York, NY: W.H. Freeman, 1990. ISBN: 0716720310. (Out of print.)

There are two other recommended books for this course:

Amazon logo Hoffman, K., and R. Kunze. Linear Algebra. 2nd ed. Upper Saddle River, NJ: Prentice Hall, 1971. ISBN: 0135367972.

Amazon logo Bretscher, O. Linear Algebra with Applications. 3rd ed. Upper Saddle River, NJ: Prentice Hall, 2004. ISBN: 0131453343.


Lec #TOPICSReadings
1Systems of Linear EquationsSection 0.1: Systems of linear equations, row equivalence
2Echelon FormSection 0.2: Gaussian and Gauss-Jordan elimination, (reduced) row-echelon form, back-substitution
3MatricesSection 0.3: Matrices, matrix operations, block multiplication
4Matrices (cont.)Section 0.4: Matrices and linear systems, elementary matrices, (reduced) row-echelon matrices
5Solution SpacesSection 0.5: The space of solutions to a homogeneous linear system, uniqueness of the reduced row-echelon form, matrix rank, criterion for existence of solutions
6Inverses and TransposesSection 0.6: Matrix inverses (right, left), invertible matrices, transpose of a matrix, symmetric matrices
7Fields and SpansSection 1.1: Definition of a field F, examples: Q, R, C, Z/pZ (see also 1.6. pp. 132-133), linear combinations of vectors, and spans in Fn
8Vector SpacesSection 1.2: Vector spaces, definition and examples, sub-spaces, the row space, column space, and nullspace of a matrix
9Linear IndependenceSection 1.3: Linear independent vectors
10Basis and DimensionSection 1.4: Basis of a vector space, dimension, bases for the row space and column space of a matrix, Rank plus nullity theorem for matrices, Basis extension theorem
11CoordinatesSection 1.5: Coordinates with respect to an ordered basis, change of coordinates matrix
12Review for Quiz 1
13Quiz 1 (Chapters 0-1)
14DeterminantsSection 2.1 (pp. 137-143): Determinant function (definition, properties, uniqueness), computing determinants using row-reduction

Section 2.1 (pp. 144-146): invertible matrices, det(AB) = det(A)det(B), det(At) = det (A)
15PermutationsSection 2.2: Permutations and the permutation definition of the determinant
16Determinants (cont.)Section 2.2: Permutations and the permutation definition of the determinant
17Laplace ExpansionSection 2.3: Cofactor (Laplace) expansion of the determinant, the adjoint of a matrix, finding the inverse using the adjoint, Cramer's rule
18Review for Quiz 2
19Quiz 2 (Chapter 2)
20Linear TransformationsSection 3.1: Linear transformations (definition, examples), matrix associated to a linear transformation
21Rank, Kernel, ImageSection 3.2: Properties of linear transformations, rank, kernel, image, Rank plus nullity theorem for linear transformations, one-one, onto, isomorphism
22Matrix RepresentationsSection 3.3: Matrix representations for linear transformations, similar matrices
23EigenspacesSection 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial
24Eigenspaces (cont.)Section 3.4: Eigenvalues, eigenvectors (definitions and examples), eigenspaces, characteristic polynomial
25DiagonalizationSection 3.5: Diagonalizable linear operators and matrices
26Cayley-Hamilton TheoremSection 6.1: Cayley-Hamilton theorem, minimal polynomial
27Jordan Canonical FormSection 6.4 (pp. 373-376): Jordan form, generalized eigenvectors and Primary decomposition theorem from section 6.5 (see also J. Starr's notes from Fall 2004)
28Review for Quiz 3
29Quiz 3 (Chapter 3)
30Computing Generalized EigenvectorsSection 6.4 (pp. 376-384): More on Jordan form, computing generalized eigenvectors
31Norms and Inner ProductsSection 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality
32Norms and Inner Products (cont.)Section 4.1: Norms, real and hermitian inner products (definitions, examples), projections, Schwartz' inequality
33Orthogonal BasesSection 4.2: Orthogonal and orthonormal bases, Gram-Schmidt algorithm, QR decomposition
34Orthogonal ProjectionsSection 4.3: Orthogonal projections, orthogonal complement, direct sums
35Isometries, Spectral TheorySection 4.5 (pp. 282-285): Isometries, orthogonal and unitary matrices
36Singular Value DecompositionSection 4.5 (pp. 286-291): Self-adjoint operators, symmetric and hermitian matrices, eigenvalues of self-adjoint operators, Principal axis theorem, Spectral resolution
37Polar DecompositionSection 4.6: Singular value decomposition, positive (semi)definite matrices, Polar decomposition
38Review for the Final

 








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